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A Centuries-Old, Higher-Dimensional Math Problem Was Just Solved

“It’s stunningly simple.”

Mathematicians have been studying how to stack spheres since the 17th century, when Johannes Kepler hypothesized that the densest way to pack together equally-sized spheres in space is the familiar pyramid piling we see of apples and oranges in grocery stores.

Despite the problem seeming rather simplistic (and trust me, it’s not), the debate wasn’t settled until 1998, when Thomas Hales, now of the University of Pittsburgh, proved Kepler’s conjecture in 250 pages of mathematics and computer calculations.

Now, a Ukrainian mathematician, Maryna Viazovska, has solved two versions of this centuries-old problem in higher dimensions. She has proved that the densest possible way to pack spheres together is possible in the 8th and 24th dimensions.

Higher-dimensional sphere packings are extremely difficult to visualize — it’s easier to think of them as a set of coordinates. In three dimensions, a sphere is defined a set of points that are all the same distance from a common center, given as the coordinates x, y, z (length, width, height). Similarly, in higher-dimensional space, all the points that define a sphere are also at a fixed distance away from the center point, but the coordinates have a variable for each new dimension: w, x, y, z, etc.

All of us have seen a pile of tennis or golf balls before, so you know that there is always some empty space between the balls. For example, if the balls are just poured into a bag, about 36 percent of the resulting pile will be air. However, that number drops to 26 percent if you arrange the spheres precisely. This is called the 26 Percent Method, and mathematicians have known that in three-dimensions, there is no better way to do it.

But that is not necessarily true for higher dimensions. Although it would make sense that the more dimensions there are, the more complicated the spheres are to stack, mathematicians have long known that there are two very special dimensions: eight (E8) and 24 (Leech lattice). These dimensions pack spheres better than all the other dimensions known to mathematicians.

Viazovska started off with an arrangement of eight-dimensional spheres. E8 looks a lot like a higher-dimensional version of the 26 Percent Method, except that in eight dimensions, there is enough space between the spheres that a new one can fit snugly between them. Amazingly, she was able to show that E8 leaves no extra space anywhere — it is the most efficient way of stacking eight-dimensional spheres together. Proving mathematicians’ age-old theory correct.

After Viazovska published her paper online at arXiv.org, she was contacted by other mathematicians who wanted her assistance in solving the same problem in 24 dimensions.

The Leech lattice was discovered by John Leech in the 1960s. He determined a way of arranging data in 24 dimensions — just like how someone might arrange spheres — that made it very robust for tasks like transmitting pictures of Jupiter from half-a-billion miles away, which Voyagers 1 and 2 did a decade later. And mathematicians believed that this was the most efficient way to do it.

Once again, Viazovska and her collaborators proved it correct in a paper that was published just one week after her E8 paper.

Although the two papers haven’t been peer-reviewed yet, there is little doubt among mathematicians that they’re right. “It’s stunningly simple, as all great things are,” said Peter Sarnak, of Princeton University and the Institute for Advanced Study, to Quanta Magazine. “You just start reading the paper and you know this is correct.”

Since E8 and the Leech lattice are connected to so many areas of mathematics and physics, Viazovska and her group’s approach will likely lead to several more discoveries and future proofs.